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Simple charge diffusion model

Simple charge diffusion model. Paul Dauncey, Anne-Marie Magnan. Epitaxial layer charge movement. Modelled in detail by Giulio All effects included Detailed geometry Hard to get intuitive feel for results Difficult to run many variations due to speed restrictions

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Simple charge diffusion model

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  1. Simple charge diffusion model Paul Dauncey, Anne-Marie Magnan Paul Dauncey

  2. Epitaxial layer charge movement • Modelled in detail by Giulio • All effects included • Detailed geometry • Hard to get intuitive feel for results • Difficult to run many variations due to speed restrictions • Wanted to see how much due to diffusion • Assumption that this is dominant movement mechanism • Make simple model for diffusion • Numerically solve and compare to Giulio’s more detailed simulation • NOT a replacement for Giulio’s work • Need to calibrate to his results to set scale • But allows quick interpolation and test of other geometries, charge deposits, etc. Paul Dauncey

  3. Diffusion model • Basic equations • Charge conservation: dr/dt + .j = 0 (so no recombination) • Diffusive movement: j = −kr where k is the diffusion constant • These can be combined to give dr/d(kt) = 2r • Time scaled by k, so no absolute timescale • Work with 5×5 pixel grid • 20×20 points per pixel, each 2.5×2.5mm2 • Divide epitaxial depth with same cell size • 15mm/2.5mm = 6 cells • Use very simple numerics • Three-point O(Dx2) for 2 • Forward (Newton) O(kDt) time step • Boundary conditions a bit tricky • Perfect boundary at bottom of epitaxial layer • Fraction of charge removed for some cells at top of epitaxial layer • Exponential falloff through 5×5 pixel grid edges Paul Dauncey

  4. First run with no n-well Magnitude is rate of charge absorbed in top cells; values chosen by comparing with Giulio’s results Paul Dauncey

  5. Charge diffusion movie 18 Jan 2009 Paul Dauncey Paul Dauncey 5

  6. Time dependence • Want final results after charge has been collected/diffused out • Note, Giulio reports 90% charge levels so some differences • Worst-case for time as least charge absorbed by pixels • Total = 1.0, Bulk → 0.0, 3×3 pixels ~ 0.95, Central pixel ~ 0.45, Outside 5×5 < 0.01 18 Jan 2009 Paul Dauncey Paul Dauncey 6

  7. Overall final distribution • Central pixel ~ 0.45 • Side pixel ~ 0.1 • Corner pixel ~ 0.03 Paul Dauncey

  8. Repeat for the usual 21 points 8 9 7 5 4 6 2 3 1 Paul Dauncey

  9. Comparison with Giulio’s results Giulio Diffusion Paul Dauncey

  10. Add n-well with no deep p-well New n-wells Original diodes Paul Dauncey

  11. Comparison with Giulio’s results Giulio Diffusion 18 Jan 2009 Paul Dauncey Paul Dauncey 11

  12. Add deep p-well N-wells absorption reduced by a factor of 250 18 Jan 2009 Paul Dauncey Paul Dauncey 12

  13. Comparison with Giulio’s results Giulio Diffusion 18 Jan 2009 Paul Dauncey Paul Dauncey 13

  14. Just model central n-wells N-wells without deep p-well but only in central pixel 18 Jan 2009 Paul Dauncey Paul Dauncey 14

  15. Comparison with Giulio’s results Giulio Diffusion 18 Jan 2009 Paul Dauncey Paul Dauncey 15

  16. Conclusions • Main effect seems to be diffusion • Modelling with simple simulation is reasonable but quantitatively there is disagreement • Ideal or realistic deep p-well agrees to within ~50% • No deep p-well differs by order of magnitude in tails • Can help quickly to quantify differences • Missing n-wells outside central pixel • Charge lost outside 3×3 pixels, etc. • Not a substitute for full simulation 18 Jan 2009 Paul Dauncey Paul Dauncey 16

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